Magnetic force microscope

The magnetic force microscope (MFM) is a variety of atomic force microscope, where a sharp magnetized tip scans a magnetic sample; the tip-sample magnetic interactions are detected and used to reconstruct the magnetic structure of the sample surface. Many kinds of magnetic interactions are measured by MFM, including magnetic dipole–dipole interaction. MFM scanning often uses non-contact AFM (NC-AFM) mode.

Because the stray magnetic field from the sample can affect the magnetic state of the tip, and vice versa, interpretation of the MFM measurement is not straightforward. For instance, the geometry of the tip magnetization must be known for quantitative analysis.

Typical resolution of 30 nm can be achieved,[3] although resolutions as low as 10 to 20 nm are attainable.[4]

Nowadays, tips are batch fabricated (tip-cantilever) using a combination of micromachining and photolithography. As a result, smaller tips are possible, and better mechanical control of the tip-cantilever is obtained.[11][12][13]

Cantilever can be made of single-crystalline silicon, silicon dioxide (SiO2), or silicon nitride (Si3N4). The Si3N4 cantilever-tip modules are usually more durable and have smaller restoring force constants (k).

Tips are coated with a thin (< 50 nm) magnetic film (such as Ni or Co), usually of high coercivity, so that the tip magnetic state (or magnetization M) does not change during the imaging.

The tip-cantilever module is driven close to the resonance frequency by a piezoelectric crystal with typical frequencies ranging from 10 kHz to 1 MHz.[5]

The scanning method when using an MFM is called the "lift height" method.[14] When the tip scans the surface of a sample at close distances (< 10 nm), not only magnetic forces are sensed, but also atomic and electrostatic forces. The lift height method helps to enhance the magnetic contrast through the following:

First, the topographic profile of each scan line is measured. That is, the tip is brought into a close proximity of the sample to take AFM measurements.

For small deflections, the tip-cantilever can be modeled as a damped harmonic oscillator with a proof mass (m) in [kg], an ideal spring constant (k) in [N/m], and a damper (D) in [N·s/m].[16]

If an external oscillating force Fz is applied to the cantilever, then the tip will be displaced by an amount z. Moreover, the displacement will also harmonically oscillate, but with a phase shift between applied force and displacement given by:[5][6][9]

Dynamic mode of operation refers to measurements of the shifts in the resonance frequency.

The cantilever is driven to its resonance frequency and frequency shifts are detected.

Assuming small vibration amplitudes (which is generally true in MFM measurements), to a first-order approximation, the resonance frequency can be related to the natural frequency and the force gradient. That is, the shift in the resonance frequency is a result of changes in the spring constant due to the (repelling and attraction) forces acting on the tip.

The change in the natural resonance frequency is given by

, where

For instance, the coordinate system is such that positive z is away from or perpendicular to the sample surface, so that an attractive force would be in the negative direction (F<0), and thus the gradient is positive. Consequently, for attractive forces, the resonance frequency of the cantilever decreases (as described by the equation). The image is encoded in such a way that attractive forces are generally depicted in black color, while repelling forces are coded white.

Theoretically, the magneto-static energy (U) of the tip-sample system can be calculated in one of two ways:[1][5][6][17]

One can either compute the magnetization (M) of the tip in the presence of the magnetic stray field (H) of the sample or

Compute the magnetization of the sample in the presence of the magnetic stray field of the tip (whichever is easier)

Then, integrate the (dot) product of the magnetization and stray field over the interaction volume as

and compute the gradient of the energy over distance to obtain the force F. Assuming that the cantilever deflects along the z-axis, and the tip is magnetized along a certain direction (e.g. the z-axis), then the equations can be simplified to

Since the tip is magnetized along a specific direction, it will be sensitive to the component of the magnetic stray field of the sample which is aligned to the same direction.

The MFM can be used to image various magnetic structures including domain walls (Bloch and Neel), closure domains, recorded magnetic bits, etc. Furthermore, motion of domain wall can also be studied in an external magnetic field. MFM images of various materials can be seen in the following books and journal publications:[5][6][18] thin films, nanoparticles, nanowires, permalloy disks and recording media.

There have been several attempts to overcome the limitations mentioned above and to improve the resolution limits of MFM. For example, the limitations from air flow has been overcome by MFMs that operate at vacuum.[19] The tip-sample effects have been understood and solved by several approaches. Wu et al., have used a tip with antiferromagnetically coupled magnetic layers in an attempt to produce a dipole only at the apex.[20]